diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index f9318fb..fa8f765 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -62,9 +62,37 @@ "icon": "lucide-file", "title": "Trigonometri" } + }, + { + "id": "be47d5ede3a9176b", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Komplexa tal.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Komplexa tal" + } + }, + { + "id": "76c8d943958d45bf", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Gräsvärde (1).md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Gräsvärde (1)" + } } ], - "currentTab": 3 + "currentTab": 5 } ], "direction": "vertical" @@ -224,12 +252,17 @@ "obsidian-git:Open Git source control": false } }, - "active": "4eef5f8feb086f9e", + "active": "76c8d943958d45bf", "lastOpenFiles": [ - "Funktioner Forts.md", - "Trigonometri.md", - "Grafer.md", + "gv1.png", "Funktioner.md", + "Funktioner Forts.md", + "Gräsvärde (1).md", + "Komplexa tal.md", + "Grafer.md", + "Trigonometri.md", + "k2.png", + "k1.png", "f_inverse.png", "g2.png", "g1.png", diff --git a/Gräsvärde (1).md b/Gräsvärde (1).md new file mode 100644 index 0000000..fbe3ef1 --- /dev/null +++ b/Gräsvärde (1).md @@ -0,0 +1,21 @@ +- Gränsvärden + - **Def**: *Om för varje $\epsilon>0$ existerar $\delta>0$ så att $$\mid{x-a}\mid<\delta\Rightarrow\mid{f(x)-L}\mid<\epsilon$$är talet $L$ gransvärde till $f(x)$ då $x$ får mot $a$. Betekning: $f(x)\longrightarrow{L}$ då $x\longrightarrow{a}$, eller $$\lim_{x\to{a}} f(x)=L$$* + - **Def**: *Om för varje $\epsilon>0$ existerar $M>0$ så att$$x>M\;\Rightarrow\;\mid{f(x)-L}\mid<\epsilon$$är talet $L$ gränsvärde till $f(x) då $x$ går mot oändlighit. Beteckning: $f(x)\longrightarrow{L}$ då $x\longrightarrow\infty$, eller $$\lim_{x\to\infty}f(x)=L$$* +- Remarks + - *Om det inte fins sådant $L$ värde, saknar funktionen gränsvärde på punkten $a$,* + - **Ex**: $$\begin{align}f(x)=\sin x\\\lim_{x\to\infty}\sin x\\\text{Existerar inte}\end{align}$$ + - **Ex**: $$\begin{align}f(x)=\sin\frac1x\\\lim_{x\to0}\sin\frac1x\\\text{Existerar inte}\end{align}$$ + - *Punkten $a$ behöver inte vara i $D_f$.* + - *Beteende av funktionen kring "problempunkter" är intressant.* + - *Långsiktig beteende hos funktioner: $$\lim_{x\longrightarrow\infty}f(x)$$* + - *Derivator, integraler, asymptot etc definieras med hjälp av gränsvärde.* + - *Om $a$ int är "problempunkt" stoppar vi in $x=a$ i $f(x)$* + - **Def**: *"Problempunkt" t.ex $\lim_{x\to 0}\frac1x$ går inte att direkt lösa på grund av division med $0$* +- One sided limits + - ![[gv1.png]] +- Problem fall + - $\left[\frac00\right]$ form: **Ex**: $$\lim_{x\to1}\frac{x^2-3x+2}{x^2-1},\;\lim_{x\to0}\frac{e^x-1}x,\;\lim_{x\to\infty}\frac{\tan{x}}x$$ + - $\left[\frac\infty\infty\right]$ form: **Ex**: $$\lim_{x\to\infty}\frac{x^2-3x+2}{x^2-1},\;\lim_{x\to\infty}\frac{x^3}{2^x}$$ + - $\left[0\times\infty\right]$ form: **Ex**: $$\lim_{x\to\infty}x^2\ln\mid{x}\mid$$ + - $\left[0^0\right]$ form: **Ex**: $$\lim_{x\to0+}x^x$$ + - $\left[\infty^0\right]$ form **Ex**: $$\lim_{x\to\infty}$$ \ No newline at end of file diff --git a/Komplexa tal.md b/Komplexa tal.md new file mode 100644 index 0000000..6d67d59 --- /dev/null +++ b/Komplexa tal.md @@ -0,0 +1,30 @@ +- Komplexa tal + - **Def**: $x^2+1=0$ saknar reell lösning. Vi antar talet $i\notin\mathbb{R}$ löser ekvationen, d.v.s $i^2=-1$ + - Mängd av komplexa talen: $\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$ + - Om $z=a+bi,a=Re(z)$ och $b=Im(z)$ + - **Konjugat**: $z=a+bi\Rightarrow\bar{z}=a-bi$ + - **Regler**: + - $\bar{\bar{z}}=z$ + - $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$ + - $\overline{z_1\times{z_2}}=\overline{z_1}\times{z_2}$ + - **Absolut belopp**: $$\mid{z}\mid=\mid\overline{z}\mid=\sqrt{z\overline{z}}=\sqrt{a^2+b^2}\text{ om }z=a+bi$$ + - **Triangelsformeln**: $\mid{z_1+z_2}\mid\leq\mid{z_1}\mid+\mid{z_2}\mid$ + - **Ex**: $$\begin{align}z_1=2+3i\\z_2=2-i\\\\z_1+z_2=(2+3i)+(2-1)\\=4+2i\\\overline{z_1+z_2}=4-2i\\\overline{z_1}=2-3i,\;\overline{z_2}=2+i\\\overline{z_1}+\overline{z_2}=2-3i+2+i\\=3-2i\\\\z_1\times{z_2}=(2+3i)(2-i)\\=4-2i+6i-3i^2\\=4+4i+3\\=7+4i\\\overline{z_1\times{z_2}}=7-4i\\\overline{z_1}=2-3i,\;\overline{z_2}=2+i\\\overline{z_1}\times\overline{z_2}=(2-3i(2+i)\\=4+2i-6i-3i^2\\=4-2i+3\\=7-4i\end{align}$$ + - **Ex 2**: $$\begin{align}z=a+bi\\\overline{z}=a-bi\\z\times\overline{z}=(a+bi)(a-bi)\\=a^2-\left(bi\right)^2\\=a^2-b^2i^2\\=a^2+b^2\end{align}$$ + - **Ex 3**: $$\begin{align}\mid{z_1+z_2}\mid=\mid4+2i\mid\\=\sqrt{4^2+2^2}\\=\sqrt{16+4}=2\sqrt{5}\\\mid{z_1}\mid=\mid2+3i=\sqrt{2^2+3^2}\\=\sqrt{13}\\\mid{z_2}\mid=\mid2-i\mid=\sqrt{2^2+(-i)^2}=\sqrt{5}\end{align}$$ + - **Ex 4**: $$\begin{align}\frac{z_1}{z_2}=\frac{2+3i}{2-i}\\=\frac{2+3i}{2-i}\times\frac{2+i}{2+i}\\=\frac{4+2i+6i+3i^2}{2^2-i^2}\\=\frac{1+8i}{5}\\=\frac{1}{5}+\frac{8}{5}i\end{align}$$ +- Grafer + - ![[k1.png]] + - ![[k2.png]] +- Polär form + - **Eulers formel**: $e^{i\theta}=\cos\theta+i\sin\theta$ + - Varje komplex tal $z=x+yi$ kan skrivas på pol'r form som $$z=re^{i\theta}$$ där $$r=\sqrt{x^2+y^2}$$ och $arg(z)=\theta$ är så att $$\cos\theta=\frac{x}{\sqrt{x^2+y^2}}\text{ och }\sin\theta=\frac{y}{x^2+y^2}$$ + - **de Moivre**: $z=re^{i\theta}\Rightarrow z^n=r^ne^{in\theta}=r^n(\cos(n\theta)+i\sin(n\theta))$ + - **Ex**: Lös $z^3=1+i\sqrt3$ $$\begin{align*}1+i\sqrt3=n_\circ e^{i\theta}, \theta\in\left[0,2\pi\right)\\n_\circ=\sqrt{1^2+\left(\sqrt3\right)^2}=2\\\theta\in\left[0,2\pi\right)\text{ uppfyller}\\\cos\theta=\frac12,\sin\theta=\frac{\sqrt3}2\\\Rightarrow\theta=\frac\pi3\\z^3=1+i\sqrt3=2e^{i\frac\pi3}\\\text{Låt }z=n_1e^{i\phi}\\\text{Då är }z^3=n_1^3e^{i3\phi}\\\Leftrightarrow\left\{\begin{aligned}n_1^3=2,n\in\mathbb{R}\\e\phi=\frac\pi3+2\pi{k},k\in\mathbb{Z},\phi\in\left[0,2\pi\right)\end{aligned}\right.\\\Leftrightarrow\left\{\begin{aligned}n_1=\sqrt[3]{2}\\\phi=\frac\pi9+\frac{2\pi k}{3},k=0,1,2\end{aligned}\right.\\k=0:\;\phi=\frac\pi9+\frac{2\pi}3\times0=\frac\pi9\\k=1:\;\phi=\frac\pi9+\frac{2\pi}3=\frac{7\pi}9\\k=2:\;\phi=\frac\pi9+\frac{2\pi}4\times2=\frac{13\pi}9\end{align*}$$ + - **Ex 2**: $$\begin{align}z=-\frac{\sqrt3}{2}+\frac{1}{2}i\\z=ne^{i\theta}\\n=\sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2}=1\\\theta\text{ är så att }\cos\theta=\frac{\sqrt{3}}{2},\sin\theta=\frac{1}{2}\\\text{En lösning}:\theta=\pi-\frac{\pi}{6}=\frac{5\pi}{6}\\\text{Alla lösningar}:\theta=\frac{5\pi}{6}+2\pi{n},n\in\mathbb{Z}\\z=e^{i\left(\frac{5\pi}{6}+2\pi{n}\right)},n\in\mathbb{Z}\\\text{Svar: }z=e^{i\frac{5\pi}{6}}\end{align}$$ +- Polynom + - **Theorem**: *Algebrans huvudsats: Polynomet$$p(z)=c_nz^n+c_{n-1}z^{n-1}\dots+c_0,\;c_k\in\mathbb{C}$$har en rot i $\mathbb{C}$. D.v.s det finns en $z_1\in\mathbb{C}$ så att $p(z_1)=0$.* + - **Faktorsats**: $p(z)=(z-z_1)q(z)$ + - **Theorem**: *Polynomet ovan kan skrivas som $p(z)=c_n(z-z_1)\dots(z-z_n)$. Alla polynom har $n$ komplexa rötter (och faktorer).* + - **Theorem**: *Polynom med reella koefficienter:$p(x)=a_nz^n+a_{n-1}z^{n-1}\dots+a_0,\;a_k\in\mathbb{R}$. Om $z_0$ är en rot så är $\overline{z_0}$* + - **Ex**: $$\begin{align}p(z)=3z^3-7z^2+17z-5\\p(1+2i)=0\\\text{Polynomet har reella koefficienten. även konjugatet 1-2i är en rot.}\\\text{Enlight faktorsatsen}\\p(z)=(z-1-2i)(z-1+2i)q(z)\\\text{för något polynom }q(z)\\p(z)=\left(\left(z-1\right)^2-\left(2i\right)^2\right)q(z)\\=\left(z^2-2z+1+4\right)q(z)\\=\left(z^2-2z+5\right)q(z)\\\text{Polynomdivision: }\\\frac{3z-1}{z^2-2z+5}\\p(z)=\left(z-1-2i\right)\left(z-1+2i\right)\left(3z-1\right)\\\text{Rötter: }1+2i,1-2i,\frac13\end{align}$$ \ No newline at end of file diff --git a/Trigonometri.md b/Trigonometri.md index a75978f..8fb2176 100644 --- a/Trigonometri.md +++ b/Trigonometri.md @@ -26,4 +26,13 @@ - $\sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.$ - $\cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.$ - $\tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}$ - - Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ \ No newline at end of file + - Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ +- Inverse trigonometric function + - **Def**: *$f(x)=\sin(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$. Then $f$ is strictly increasing on $D_f$ and hence inverible. The fuction $\arcsin$ is defined as $$\arcsin(x)=f^{-1}(x)\text{ on }D_{arcsin}=R_f=[-1,1]$$* + - **Similarly**: *For $g(x)=\cos(x),x\in\left[0,\pi\right]$ (which is strictly decreasing) and $h(x)=\tan(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ (which is strictly increasing), the function $\arccos$ and $\arctan$ are defined as $$\begin{align}\arccos(x)=g^{-1}(x)\text{ on }D_{\arccos}=\left[-1,1\right]\\\arctan(x)=h^{-1}(x)\text{ on }D_{\arctan}=\mathbb{R}\end{align}$$* + - **Note**: *That the tanges $R_{\arcsin}=R_{\arctan}=\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$ whereas $R_{\arccos}=\left[0,\pi\right]$* +- Properties + - **Def**: $$\begin{align}\sin(\arcsin(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arcsin(sin(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\\cos(\arccos(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arccos(\cos(x))=x\text{ if }x\in\left[0,\pi\right]\\\tan(\arctan(x))=x\forall{x}\in\mathbb{R}\text{ | }\arctan(\tan(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\end{align}$$ + - **Complementary angles**: $$\arcsin(x)+\arccos(x)=\frac{pi}{2},\;\arctan(x)+\arccos(x)=\frac{\pi}{2}$$ + - **Negatives**: *$\arcsin$ and $\arctan$ are odd functions. $$\begin{align}\arcsin(-x)=-\arcsin(x)\\\arccos(-x)=\pi-\arccos(x)\\\arctan(-x)=-\arctan(x)\end{align}$$* +- \ No newline at end of file diff --git a/gv1.png b/gv1.png new file mode 100644 index 0000000..0e2aaab Binary files /dev/null and b/gv1.png differ diff --git a/k1.png b/k1.png new file mode 100644 index 0000000..09d92a1 Binary files /dev/null and b/k1.png differ diff --git a/k2.png b/k2.png new file mode 100644 index 0000000..52d033d Binary files /dev/null and b/k2.png differ